The Bethe Ansatz Equations for Reflecting Magnons

We derive the Bethe Ansatz Equations on the half line for particles interacting through factorized $S$-matrices invariant relative to the centrally extended $su(2|2)$ Lie superalgebra and $su(1|2)$ open boundaries. These equations may be of relevance for the study of the spectrum of open strings on $AdS_{5} \times S^5$ background attached to Y=0 giant graviton branes. An one-dimensional spin chain hamiltonian associated to this system is also derived.